Visualizing Dessins D’Enfants - Willamette...
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IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Visualizing Dessins D’EnfantsWillamette Valley Consortium for Mathematics Research
Mary Kemp and Susan Maslak
Occidental College and Ave Maria University
MAA MathFestAugust 7, 2014
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
IntroductionBelyi MapsDessinsPassportsShabat Polynomials
Finding Belyi MapsQuestionExampleGeneral Results
The Pell-Abel Equation: An Application
Closing Remarks
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Belyi Maps
DefinitionA Belyi Map is a function F : X → C with the critical values inthe set {0, 1,∞}
I X is a Riemann SurfaceI C = C ∪ {∞} is equivalent to a sphere in R3
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
DefinitionA Dessin D’Enfant, or Dessin for short, is a connected bicoloredgraph that is embedded on a Riemann surface and is obtained by aBelyi function, f , in the following way:
I F−1(0)→ black vertices
I F−1(1)→ white vertices (shown as red in our pictures)
I F−1([0, 1])→ edges
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Examples of Dessins and Their Belyi Maps
0.1 0.2 0.3 0.4 0.5 0.6Re�z�
�0.4
�0.2
0.2
0.4
Im�z� z � �3125 �4 z � 1�6186 624 z
�1.0 �0.5 0.5 1.0Re�z�
�1.5
�1.0
�0.5
0.5
1.0
1.5Im�z�z �
�1� z�3 �35 z4 � 40 z3 � 48 z2 � 64 z � 128�64 398046511104
−3125(4z−1)6
186624z(1−z)3(35z4+40z3+48z2+64z+128)6
4398046511104
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Passports
DefinitionA Passport is a set of valencies (or degrees) that corresponds tothe vertices of the dessin, which is represented in the form[b1, b2, ..., bn; w1,w2, ...,wk ] where b1....bn are the the blackdegrees and w1, ...,wk are the white degrees.
Example
[5, 4, 3; 3, 1, 1, 1, 1, 1, 1, 1, 1, 1] can also be written as [5, 4, 3; 3, 19]
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Shabat Polynomials
I A Shabat Polynomial refers to a Belyi Map defined byF : C 7→ C, which is represented by a tree (a graph with nocycles).
I given a tree we can find its Shabat polynomial by solving:I F (z) = 0 → black verticesI F (z)− 1 = 0 → white vertices
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Finding Belyi Maps
QuestionCan we find all Belyi maps for a given passport of size k?
[33, 15; 27]→
-1.0 -0.5 0.5 1.0ReHzL
-2
-1
1
2
ImHzLz ®
4 H1 - zL z3 I2 z
2 + 3 z + 9M3 I8 z4 + 28 z
3 + 126 z2 + 189 z + 378M
531 441
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Example
The passport [s2, r 2; 4, 1(r−1)2+(s−1)2] is size k = 2.
[42, 22; 4, 18]
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
F (z) = A(z − c0)4(z − c1)4(z − c2)2(z − c3)2
F (z)− 1 = A(z − d0)4(z8 + d8z7 + d7z
6 + · · ·+ d2z + d1)
[42, 22; 4, 18]
Belyi map when s = 4, r = 2:
F (z) = 1256 (2z2 + 4)4(z2 − 1)2
General Form:F (z) = s−s(−1)r (z2−1)r (s + rz2)s
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Belyi Maps for Passports with only one Dessin (k=1 inShabat-Zvonkin paper)
Passport Belyi Map[n; 1n] zn
[2n, 1; 2n, 1]1 + cos ((2n + 1) arccos(z))
2
[2n; 2n−1, 12]1 + cos (2n arccos(z))
2
[sr−1, t; r , 1(r−1)(s−1)+(t−1)] (1− z)t
(r−1∑k=0
( t
s
)k
zk
k!
)s
[r , t, 1r+t−2; 2r+t−1] 4Sr ,t(z)(1− Sr ,t(z))
[n2, 14n−3; 32n−1] −3√
3 i Sr (z) (1− Sr (z))(
Sr (z)− 1−i√
32
)[33, 15; 27] − 4
531441 (z − 1)z3(2z2 + 3z + 9
)3 (8z4 + 28z3 + 126z2 + 189z + 378
)Where Sr,t (z) = (1 − z)t
r−1∑j=0
(t − 1 + j
t − 1
)z j and Sr (z) = Sr,r (z)
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Belyi Maps for passports with exactly two Dessins (k=2)
Passport Belyi Map
[r 2, s2; 4, 12r+2s−4]s−s(−1)r (z2 − 1)r (s + rz2)s
?
[r , t, 12r+2t−3); 3r+t−1]−3√
3 i Sr ,t(z) (1− Sr ,t(z))(
Sr ,t(z)− 1−i√
32
)?
[r 2, 12+3k ; 4k ]−16(−1 + Sr (z))Sr (z)(1/2− Sr (z) + Sr (z)2)
4(−1 + Sr (z))Sr (z)((−1 + i)− (1 + 2i)Sr (z) + Sr (z)2)
[r 2, 13+4k ; 5k ]
C (−1 + Sr (z))Sr (z)[i(−2 +√
5) +√
5 + 2√
5)− (−3i + 2i√
5 +√5(5− 2
√5))Sr (z) + (3i +
√5(5− 2
√5))Sr (z)2 − 2iSr (z)3]
C (−1 + Sr (z))Sr (z)(i(2 +√
5 + i√
5 + 2√
5)− (3i + 2i√
5 +√5(5 + 2
√5))Sr (z) + (−3i +
√5(5 + 2
√5))Sr (z)2 + 2iSr (z)3)
[32, 1; 22, 13]C (11i +
√7− 8iz)(−1 + z)3z3
C (3i +√
7 + 8iz)(−1 + z)3z3
[3, 22; 22, 13]Cz3(35(5+3
√21−
√70 + 30
√21)−21(17+3
√21−
√70 + 30
√21)z +180z2)2
Cz3(−35(−5 + 3√
21 + i√−70 + 30
√21) + 21(−17 + 3
√21 +
i√−70 + 30
√21)z + 180z2)2
[3, 22, 13; 25]−1
1024 (z − 1)z3(8 + 4z + 3z2)2(40 + 15z + 9z2)−4729 (z − 1)2z2(3 + 2z)3(−45− 10z + 20z2 + 8z3)
[43, 18; 210]−4(−2+z)4(−1+z)4z4(5−8z +4z2)(−1−4z−10z2−20z3+45z4−24z5+4z6)
−125389989167104 z4(36− 6z + z2)4(6 + 4z + z2)(116640− 46656z + 12960z2 −
2160z3 + 270z4 − 20z5 + z6)
Where C is a large constant (too large for the table!)
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
An Application
Let F (z) = 1256 (2z2 + 4)4(z2 − 1)2 be our equation found earlier.
-1.0 -0.5 0.5 1.0ReHzL
-1.5
-1.0
-0.5
0.5
1.0
1.5
ImHzL
z ®1
256Iz2 - 1M2 I2 z
2 + 4M4
If D(z) =∏
(z − di ) = (3 + z2)(−8 + 3z4 + z6)Then P(z) = 1
8 (−8 + 24z4 + 8z6 − 9z8 − 6z10 − z12)and Q(z) = 1
8 z2(−1 + z2)(2 + z2)2
is a solution to the equation P(z)2 − D(z)Q(z)2 = 1Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
The Pell-Abel Equation
QuestionSolve for P(z)2 − D(z)Q(z)2 = 1 for a given D(z)
TheoremLet T (z) be a Shabat Polynomial. Then there exists acorresponding D(z) of the form D(z) =
∏(z − di ), where di
represent the coordinates of the odd vertices, for whichP(z)2 − D(z)Q(z)2 = 1 has infinitely many solutions.
ExampleR(z) = s−s(−1)r (z2 − 1)r (s + rz2)s where s = 2n and r = 2
D(z) =
(n∑
k=1
(n+1k+1
)knn−k−1z2k−2
)(−2nn +
n∑k=1
(n+1k+1
)knn−k−1z2k+2
)where −2R(z) + 1 and [−2R(z)+1]2−1
D(z) is one of infinitely many solutions.
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Future Projects and Open Questions
I Inverse Enumeration for Belyi Maps with multiple faces
I Belyi Maps for Cartesian Products of graphs
I Exploration of the role Galois Action plays on Belyi Maps andDessins
I Other applications Belyi Maps have on Diophantine Equations
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
Acknowledgements
I Willamette Valley Consortium for Mathematics Research, NSF
I Dr. Naiomi Cameron
I Lewis and Clark College
I Ave Maria University and Occidental College
I Dr. Edray Goins from Purdue University
I Computer Science Whizzes
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
References
J. Couveignes, Calcul et rationalite de fonctions de Belyi en genre 0, Annales de l’Institut
Fourier (Grenoble) 44 (1994), no. 1, 1–38.
L. Granboulan, Calcul d’objets geometriques a l’aide de methodes algebraiques et
numeriques: dessins d’enfants, Ph.D. thesis, Universite Paris 7, 1997.
Y. Kochetkov, Geometry of planar trees. (Russian) Fundam. Prikl. Mat. 13 (2007), no. 6,
149–158; translation in J. Math. Sci. (N. Y.) 158 (2009), no. 1, 106113.
S. Lando, A. Zvonkin, Graphs on surfaces and their applications, Encyclopedia of
Mathematical Sciences, Low-Dimensional Topology, II, Springer-Verlag, Berlin, 2004.
Y. Matiyasevich, Computer evaluation of generalized Chebyshev polynomials, em Moscow
Univ. Math. Bull. 51 (1996), no. 6, 39–40.
G. Shabat, A. Zvonkin, Plane trees and algebraic numbers, (English summary) Jerusalem
combinatorics ’93, 233275, Contemp. Math., 178, Amer. Math. Soc., Providence, RI, 1994.
L. Schneps, The Grothendieck theory of dessins d’enfants (Luminy, 1993), London Math.
Soc. Lecture Note Ser., 200, Cambridge Univ. Press, Cambridge, 1994.
J. Sijsling, J. Voight, On computing Belyi maps, arXiv: 1311.2529v3 [math.NT], 2014.
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants
IntroductionFinding Belyi Maps
The Pell-Abel Equation: An ApplicationClosing Remarks
THANK YOU!ANY QUESTIONS?
Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants